Stopping of Protons in Copper at low velocities$^{\mathrm{\ast )}}$

An \textit{ab initio} study including non-linear effects is reported of the electronic stopping power~ of protons in copper over a wide range of proton velocities (v $=$ 0.01-10 a.u.).~Time-dependent density functional theory coupled with molecular dynamics [1] is used to study~electronic excitations produced by energetic protons. A plane-wave pseudo-potential scheme~ is employed to solve the time-dependent Kohn-Sham equations for a moving proton in a periodic~copper crystal. These electronic excitations and the band structure determine the stopping power~of the material and alter the interatomic forces for both channeling and off-channeling trajectories.~Our off-channeling results are in quantitative agreement with experiments. Details will be presented at the conference [1] E. Runge and E. R. U. Gross, Phys. Rev. Lett. \textbf{52,} 997 (1984). $^{\mathrm{\ast )}}$ Supported by NNSA